# Dependency derivations¶

In this section the formal rules are presented which are used to derive dependencies between components, variables, needs, goal-specifications, transformation-specifications, behavior-specifications design-specifications, relations, and combinations thereof.

The dependency sets presented in this section contain dependencies between elements across all levels of the decomposition tree. Hence, the presented sets are independent of the view one aims to obtain of the system. By selecting and visualizing sub-sets of the presented dependency sets one can create different views on the system.

## Graphical examples¶

The decomposition-function view, decomposition-function-behavior view, and decomposition-relation-design view show graphical representations of three separate example specifications which are used to illustrate the derivation rules. In the figures, components are represented by rectangles, variables are represented by circles, two components connect by a variable form a goal specification, two variables connected by a component form a transformation specification. Behavior specifications are represented by a diamond. Relations are represented a parallelogram connected to variables. Variable comparisons (design specifications) are represented by triangles connected to variables. The decomposition-function view illustrates the derivation of function dependencies, the decomposition-function-behavior view illustrates the derivation of behavior dependencies, and decomposition-relation-design view illustrates the derivation of design dependencies.

The figure above shows an example system decomposition tree composed of components \(c_1\), \(c_2\), and \(c_3\) at decomposition level \(k=1\). Component \(c_2\) is decomposed into components \(c_{21}\) and \(c_{22}\). In turn, component \(c_{22}\) is decomposed into components \(c_{221}\) and \(c_{222}\). Component \(c_3\) is decomposed into \(c_{31}\) and \(c_{32}\). Additionally, the example contains the graphical representation of five goal-specifications (\(g_1\) through \(g_5\)) and six transformation-specifications (\(t_1\) through \(t_6\)) indicating the transfer and transformation of five flow variables (\(v_1\) through \(v_5\)).

In the hierarchical specification view, displayed in the earlier figure, the component decomposition tree and function specifications are combined, visualizing the flow of variables through the various components and decomposition layers. That is, \(c_1\) provides \(v_1\) to \(c_2\) (\(g_1\)), that must internally transform \(v_1\) into \(v_2\) (\(t_1\)) and subsequently provide \(v_2\) to \(c_3\). Internal to \(c_2\), \(v_1\) is passed on to child component \(c_{21}\) which transforms \(v_1\) into \(v_3\) (\(t_2\)) and, subsequently, provides \(v_3\) to \(c_{22}\) (\(g_3\)), which must transform \(v_3\) into \(v_2\) (\(t_3\)). Flow \(v_3\) is passed on to child component \(c_{221}\) which transforms \(v_3\) into \(v_4\) (\(t_4\)) and, subsequently, provides \(v_4\) to \(c_{222}\) (\(g_4\)). Component \(c_{222}\) transforms \(v_{4}\) into \(v_{2}\) (\(t_{5}\)). \(v_{2}\) is passed to parent components \(c_{22}\) and \(c_2\). Component \(c_2\) provides \(v_2\) to \(c_3\) (\(g_2\)), which passes \(v_2\) on to child component \(c_{31}\). Finally, \(c_{31}\) transforms \(v_2\) into \(v_5\) (\(t_6\)) and provides \(v_5\) to \(c_{32}\). Note that the function chain \(t_2\) \(\rightarrow\) \(g_3\) \(\rightarrow\) \(t_3\) is a more detailed representation of transformation \(t_1\). Similarly, the function chain \(t_4\) \(\rightarrow\) \(g_4\) \(\rightarrow\) \(t_5\) is a more detailed description of transformation \(t_3\).

The figure above shows three components \(c_1\), \(c_2\), and \(c_3\), where \(c_1\) must provide \(v_1\) and \(v_2\) to \(c_2\) (\(g_1\) and \(g_2\) respectively) and \(c_2\) must provide \(v_3\) to \(c_3\). Component \(c_2\) must internally convert \(v_1\) into \(v_3\), where the value of \(v_3\) is defined by behavior specification \(h_1\) with \(v_2\) as input.

The figure above presents a subsection of the decomposition tree shown in the decomposition-function view, the graphical representation of a relation specification (\(r_1\)) and two design specifications (\(q_1\) and \(q_2\)), which are combined into a single nested specification view. In this nested view the mapping of actual variables to formal parameters is shown. For example, actual variable \(v_8\), which 'lives' in world component \(c_\mathrm{w}\), is mapped onto formal parameters \(v_8\) that'live' in components \(c_1\), \(c_2\), and \(c_{21}\), respectively. Relation \(r_1\) relates variables \(v_5\), \(v_6\), and \(v_7\) to one another. Design-specifications \(q_1\) binds the value of \(v_8\) with the value of \(v_9\) and design-specification \(q_2\) binds the value of \(v_9\) with the value of \(v_{10}\). Variables \(v_8\) and \(v_9\) are marked as being properties of components \(c_{21}\) and \(c_{211}\), respectively.

## Nomenclature¶

For compact notation, we define the following nomenclature. Let \(\mathcal{C}\) be the set of components, \(\mathcal{V}\) the set of variables, \(\mathcal{K}\) the set of variables types, \(\mathcal{N}\) the set of needs, \(\mathcal{G}\) the set of goal function specifications, \(\mathcal{T}\) the set of transformation function specifications, \(\mathcal{F} = \mathcal{G} \cup \mathcal{T}\), the set of all function specifications, \(\mathcal{H}\) the set of all behavior specifications, \(\mathcal{D}\) the set of design specifications, \(\mathcal{Q}\) the set of comparison-rules, \(\mathcal{S}\) the set of subclauses, and \(\mathcal{R}\) the set of relations specified in an instantiated ESL-specification.

An arbitrary instantiated component \(c=(V_c, P_c, N_c, C_c, G_c, T_c, H_c D_c, R_c) \in \mathcal{C}\) has variables \(V_c\subseteq \mathcal{V}\), has properties \(P_c\subseteq \mathcal{V}\) need specifications \(N_c\subseteq \mathcal{N}\), child components \(C_c\subseteq \mathcal{C}\), goal function specifications \(G_c\subseteq \mathcal{G}\), transformation function specifications \(T_c\subseteq \mathcal{T}\), behavior specification \(H_c \subseteq \mathcal{H}\), design specifications \(D_c\subseteq \mathcal{D}\), and relations \(R_c\subseteq \mathcal{R}\).

A need specification \(n= (s_n, \textit{text}_n) \in N_c\) has subject \(s_n\) and \(\textit{text}_n\) which is the unconstrained sentence specified by the user. A goal specification \(g=(c_g, c'_g, V_g, S_g)\in G_c\) has subject component \(c_g \in C_c\), indirect object component \(c'_g \in C_c\), parameters and variables \(V_g \subseteq V_c\), and sub-clauses \(S_g \subseteq \mathcal{S}\). A transformation specification \(t=(c_t, V_t, V'_t, S_t) \in T_c\), has subject component \(c_t = c\), input variables \(V_i\subseteq V_c\), output variables \(V'_i\subseteq V_c\), and sub-clauses \(S_t \subseteq \mathcal{S}\). An behavior specification \(h = \{ A_1, A_2, \ldots, A_n \} \in \mathcal{H_c}\) has \(n\) alternative cases. A case \(A = (S, S')\) is composed of a set of when clauses \(S\) and a set of then clauses \(S'\). A design specification \(d = (Q_d, S_d) \in D_c\) has design-rule set \(Q_d\) and subclauses \(S_d \subseteq \mathcal{S}\). Design-rule \(q =(v_q, b_q) \in Q_d\) has subject variable \(v_q \in V_c\) and bound \(b_q\). Bound \(b_q\) may be a variable \(v'_\mathrm{q} \in \mathcal{V}\) or a real, integer, string or boolean value. A subclause \(s = Q_m \in \mathcal{S}\) is a design-rule set \(Q_m \subseteq \mathcal{Q}\), where \(m \in \mathcal{G} \cup \mathcal{T} \cup \mathcal{D}\) is the target main clause. Design-rule \(q =(v_q, b_q) \in Q_m\) has subject variable \(v_q\) and has bound \(b_q\). A relation \(r = r = (V, V', V'') \in R_c\), where \(V \subseteq \in \mathcal{V}\) is the set of input variables, \(V' \in \mathcal{V}\) is the set of output variables and \(V'' \in \mathcal{V}\) is the set of undirected variables.

To select the components within a (sub-)branch of the decomposition tree, we define the \(tree\) operator. In an instantiated system, the set of components \(C \subseteq \mathcal{C}\) form a rooted directed tree using the parent-child relation, with world component \(c_\mathrm{w}\in C\) as root. The set of components of a (sub-)tree with a component \(c \in C\) as root is defined as:

For the example shown in \(\textit{tree}(\mathrm{c}_{2}) = \{\mathrm{c}_{2}, \mathrm{c}_{21}, \mathrm{c}_{22}, \mathrm{c}_{221}, \mathrm{c}_{222}\}\) and \(\textit{tree}(\mathrm{c}_{3}) = \{\mathrm{c}_{3}, \mathrm{c}_{31}, \mathrm{c}_{32} \}\).

To obtain the position of a component within a decomposition tree with root component \(c'\), we define the \(depth\) operator. The instantiation depth of a component \(c\in C\) with respect to a root component \(c'\) is given by:

for \(c'=(V_{c'}, P_{c'}, N_{c'}, C_{c'}, G_{c'}, T_{c'}, D_{c'}, R_{c'})\) and \(c=(V_c, P_c, N_c, C_c, G_c, T_c, H_c, D_c, R_c)\). The \(depth\) operator 'walks' down branches of the tree with root \(c'\) until it finds \(c\). For example, \(\textit{depth}(\mathrm{c}_\mathrm{w}, \mathrm{c}_{221}) = 3\), since at decomposition level \(k=1\), we have \(\mathrm{c}_2 \in C_\mathrm{c_w}\) and \(\mathrm{c}_{221} \in tree(\mathrm{c}_2)\), at decomposition level \(k=2\), we have \(\mathrm{c}_{22} \in C_\mathrm{c_{2}}\) and \(\mathrm{c}_{221} \in tree(\mathrm{c}_{22})\), at decomposition level \(k=3\) we have \(\mathrm{c}_{221} \in C_\mathrm{c_{22}}\) and \(\mathrm{c}_{221} \in tree(\mathrm{c}_{221})\), and finally, we have \(\mathrm{c}_{221} = \mathrm{c}_{221}\), yielding \(\textit{depth}(\mathrm{c}_\mathrm{w}, \mathrm{c}_{221}) = 3\). Note that, \(\textit{depth}(\mathrm{c}_{2}, \mathrm{c}_\mathrm{3}) = \mathrm{NaN}\), since \(c_3\) is not a member of the tree of \(c_{2}\).

The following sections present the derivation rules for deriving dependencies between ESL elements. All dependencies are by definition directed and represented as a tuple of source and target. For example, the dependency tuple \((a, b)\) implies that element \(b\) depends on element \(a\). If this dependency is bidirectional the tuple \((b, a)\) is added to the overall set of dependencies as well.

## Dependencies between variables¶

Three types of dependencies between variables are derived from an ESL specification: functional dependencies, i.e.,component dependencies derived from function specifications, behavior dependencies, i.e., dependencies derived from behavior specifications, and design dependencies, i.e., component dependencies derived from design specifications and from relations.

### Functional dependencies¶

A functional dependency between two variables \(v_i\) and \(v_j\) indicates that flow \(v_i\) must be transformed into flow \(v_j\) to establish the desired functionality of the system.

Transformation specifications \(\mathcal{T}\) are used to derive functional flow dependencies between variables. Variable functional flow dependency set \(E_\mathrm{v_f}\) is given by:

That is, variable \(v_j\) depends on variable \(v_i\) if \(v_i\) is a member of input variable set \(V_t\), \(v_j\) is a member of output variable set \(V_t'\), and the set of child component \(C_{c_t}\) is an empty set. In other words, only transformations of leaf components (those with no sub-components) are used to derive the functional dependencies between variables.

The derivation of this set for the example shown in decomposition-function view is straightforward, it contains all variable pairs that are connected via a transformation function fulfilled by a leaf component. That is, it equals \(\{(\mathrm{v}_{1}, \mathrm{v}_{3}), (\mathrm{v}_{3}, \mathrm{v}_{4}), (\mathrm{v}_{4}, \mathrm{v}_{2}), (\mathrm{v}_{2}, \mathrm{v}_{5}) \}\).

### Behavior dependencies¶

A behavior dependency between two variables \(v_i\) and \(v_j\) indicates that the value of a variable \(v_i\) is dynamically bound by the value of a variable \(v_j\). The set of behavior dependencies between variables \(E_{\mathrm{v_b}}\) is given by:

That is, \(E_{\mathrm{v_b}}\) consists of all variable pairs \((v, v')\) for which there exist a design-rule \((v, b)\) within design rule line \(Q\) within when-clauses \(S\) and a design-rule \((v', b')\) within design-rule-set \(Q'\) within then-clauses \(S'\). In other words, if there exists a case in which the value of \(v'\) is dynamically bound by the the value of \(v\).

For the decomposition-function-behavior view this set consists of equals \(\{(\mathrm{v}_{2}, \mathrm{v}_{3}\}\) as the value of \(\mathrm{v}_{3}\) is dynamically bound by the value of \(\mathrm{v}_{2}\) via behavior specification \(\mathrm{h}_{1}\).

### Design dependencies¶

A design dependency between two variables \(v_i\) and \(v_j\) indicates that the value of \(v_i\) is statically bound by, or directly correlated with, the value of \(v_j\). Design-rules \(\mathcal{Q}\) and relations \(\mathcal{R}\) are used to derive design dependencies between variables. Variable design dependency set \(E_\mathrm{v_d}\) is given by:

That is, two variables \(v_i\) and \(v_j\) have a dependency if there exists a design rule \((v_i, v_j) \in \mathcal{Q}\), a design rule \((v_j, v_i) \in \mathcal{Q}\), or there exists a relation \(r = (V, V', V'') \in \mathcal{R}\) such that \(v_i\) is a member of the union of the input variables \(V\) and undirected variables \(V''\) and \(v_j\) is a member of the union of the output variables \(V'\) and undirected variables \(V''\) of \(r\). Note that if \(V''\) is empty for all \(r \in \mathcal{R}\) that all dependencies between variables derived from relations become directed. Dependencies derived from design-rules always result in a bi-directional dependency.

For the decomposition-relation-design view this set equals \(\{(\mathrm{v}_{5}, \mathrm{v}_{6}), (\mathrm{v}_{7}, \mathrm{v}_{6}), (\mathrm{v}_{8}, \mathrm{v}_{9}), (\mathrm{v}_{9}, \mathrm{v}_{8}), (\mathrm{v}_{9}, \mathrm{v}_{10}), (\mathrm{v}_{10}, \mathrm{v}_{9}) \}\), where the first two pairs result from relation \(r_1\) and the last four pairs results from design rules \(q_1\) and \(q_2\), respectively.

## Dependencies between components¶

Four types of component dependencies are derived from an ESL specification: functional dependencies, i.e.,component dependencies derived from function specifications, behavior dependencies, i.e., dependencies derived from behavior specifications, design dependencies, i.e., component dependencies derived from design specifications and relations relations, and coordination dependencies, i.e. dependencies based on shared variables.

### Functional dependencies¶

Functional component dependencies are derived from the goal and transformation specifications, \(G\) and \(T\) respectively, as they define the flows through the system. The set of functional component dependencies \(E_\mathrm{c_f}\) (edges) is given by:

which is composed of four component pair sets. The first set of denotes pairs \((c_g, c'_g)\) for which subject component \(c_g\) and indirect object component \(c'_g\) are part of goal-specification \(g\), and therefore exchange a flow.

For the decomposition-function view, this set equals \(\{(\mathrm{c}_1, \mathrm{c}_2), (\mathrm{c}_2, \mathrm{c}_3), (\mathrm{c}_{21}, \mathrm{c}_{22}), (\mathrm{c}_{221}, \mathrm{c}_{222}), (\mathrm{c}_{31}, \mathrm{c}_{32}) \}\), which are all component pairs used in the respective goal function specifications. Note that the two components in such a pair are of the same decomposition level.

The second set of component pairs in the equation above denotes pairs \((c_g, c)\) for which all following conditions hold: \(c_g\) is a subject component in a goal function specification \(g\), while \(c\) is member of the tree of the indirect object component \(c'_g\) in \(g\); the tree of \(c\) contains a component \(c_t\) which is the subject of a transformation function specification \(t\); the intersection of the set of variables \(V_g\) associated with goal function \(g\) and the set of variables \(V_t\) associated with transformation function \(t\) is non-empty; and \(c'_g\) is a different component than \(c\). The component pairs that obey these conditions represent components that exist at different decomposition levels, where the second component, at a deeper decomposition level, depends on the input from the first component.

For the decomposition-function view, the pairs that obey these conditions are \(\{(\mathrm{c}_1,\mathrm{c}_{21}), (\mathrm{c}_{21}, \mathrm{c}_{221}), (\mathrm{c}_2,\mathrm{c}_{31})\}\). Consider for instance the first pair: \(\mathrm{c}_1\) provides \(\mathrm{v}_1\) to \(\mathrm{c}_2\). Component \(\mathrm{c}_2\) is the parent of \(\mathrm{c}_{21}\) and \(\mathrm{c}_{22}\), where \(\mathrm{c}_{21}\) takes \(\mathrm{v}_1\) as input to perform transformation \(\mathrm{t}_2\). Thus \(\mathrm{c}_{21}\) requires \(\mathrm{v}_{1}\) as input from \(\mathrm{c}_{1}\). Similarly, \(\mathrm{c}_{21}\) provides input to \(\mathrm{c}_{221}\).

The third set of component pairs is complementary to the second set, with similar conditions, but now from the perspective of the indirect object component \(c'_g\). This set denotes components pairs that exist at different decomposition levels but have an output flow dependency.

For the decomposition-function view, this set equals \(\{(\mathrm{c}_{222}, \mathrm{c}_{3}), (\mathrm{c}_{22}, \mathrm{c}_{3})\}\). For the first pair, the output of \(\mathrm{c}_{222}\) is required by \(\mathrm{c}_{3}\), that is \(\mathrm{c}_{222}\) carries out transformation \(\mathrm{t}_5\) to yield \(\mathrm{v}_2\) which is passed by grandparent \(\mathrm{c}_2\) to \(\mathrm{c}_3\). Similarly, viewed at the decomposition level of \(\mathrm{c}_{22}\), \(\mathrm{c}_{22}\) generates output \(\mathrm{v}_2\) that is passed on to \(\mathrm{c}_3\) by parent \(\mathrm{c}_2\).

Finally, the fourth set contains pairs \((c_i, c_j)\) for which their exist a transformation \(t\) with subject \(c_t\) which is in the tree of \(c_i\); their exist a transformation \(t'\) with subject components \(c_{t'}\) which is in the tree of \(c_j\); and the intersection of variable sets \(V'_t\) and \(V_{t'}\) or the intersection of variable sets \(V_t\) and \(V'_{t'}\) is non-empty. Additionally \(c_i\) and \(c_j\) may not be a member of each others trees. This set contains component pairs that have an input/output flow dependency based on the transformation functions they perform.

For the decomposition-function view, this set equals \(\{(\mathrm{c}_{2}, \mathrm{c}_{3}), (\mathrm{c}_{2}, \mathrm{c}_{31}), (\mathrm{c}_{22}, \mathrm{c}_{3}), (\mathrm{c}_{22}, \mathrm{c}_{31}), (\mathrm{c}_{222}, \mathrm{c}_{3}), (\mathrm{c}_{222}, \mathrm{c}_{31}), (\mathrm{c}_{221}, \mathrm{c}_{222}), (\mathrm{c}_{21}, \mathrm{c}_{221})\}\). For instance, \(\mathrm{c}_{222}\) performs transformation \(\mathrm{t}_5\) which yields \(\mathrm{v}_2\). In turn, \(\mathrm{v}_2\) is consumed by transformation \(\mathrm{t}_6\) performed by \(\mathrm{c}_{31}\), which is a child of \(\mathrm{c}_3\). This yields dependency pairs \((\mathrm{c}_{222}, \mathrm{c}_{3})\) and \((\mathrm{c}_{222}, \mathrm{c}_{31})\). Moreover, \(\mathrm{c}_{222}\) is a child of \(\mathrm{c}_{22}\) and a grandchild of \(\mathrm{c}_{2}\), which yields the additional dependency pairs \((\mathrm{c}_{22}, \mathrm{c}_{3})\), \((\mathrm{c}_{22}, \mathrm{c}_{31})\), \((\mathrm{c}_{2}, \mathrm{c}_{3})\), and \((\mathrm{c}_{2}, \mathrm{c}_{31})\).

Note that the fourth set contains dependency pairs that are also part of the first, second, or third dependency set. The reason for this is that goal-functions explicitly denote the presence of flow between components, while transformation functions implicitly denote the presence of flow between components. This redundancy in information enables us to derive the same dependencies in multiple ways and may provide us with the means to create automated specification completeness checks. For example, to check the completeness of a functional flow path between two components. These automated completeness checks are a topic of further research.

The union of the aforementioned component pair sets yields the set of functional component dependencies \(E_\mathrm{c_f}\) between components throughout the decomposition tree based on the transfer and transformation of flows.

Note

The dependency sets derived in this section are all directed. In the DSM literature, dependencies between components are usually visualized undirected. That is, product DSMs are typically symmetric. A symmetric product DSM can easily be obtained by simply duplicating the dependencies and switching the source and target.

### Behavior dependencies¶

A behavior dependency between two components \(c_i\) and \(c_j\) indicates that the dynamic functional output of component \(c_j\) dependents on the input received from component \(c_i\). The set of behavior dependencies between variables \(E_{\mathrm{v_b}}\) can used to derive the set of behavior dependencies between components \(E_{\mathrm{c_b}}\), which is given by:

wherein, the first set of component pairs denotes those pairs \(c_i\) and \(c_j\) for which there exist a goal \((c_i, c'_i, V_i, S_i) \in \mathcal{G}\) and a goal \((c_j, c'_j, V_j, S_j) \in \mathcal{G}\) such that there exists a path over the set of behavior dependencies between variables \(E_{\mathrm{v_b}}\) between a variable \(v_i\) in \(V_i\) and a variable \(v_j\) in \(V_j\), such that all intermediate variables \(v_k\) part of this path do not relate to any other transformation \(g\) or transformation \(t\). In other words, the value of a flow to be transferred by component \(c_j\) is bound by the value of a flow to be transferred by component \(c_i\), without any transformations or goals in between.

The second set of component pairs denotes those component pairs \(c_i\) and \(c_j\) for which there exist a transformation \((c_i, V_i, V'_i, S_i) \in \mathcal{T}\) and a transformation \((c_j, V_j, V'_j, S_j) \in \mathcal{T}\), such that there exists a path over the set of behavior dependencies between variables \(E_{\mathrm{v_b}}\) between a variable \(v_i\) in \(V_i\) and a variable \(v_j\) in \(V_j\), such that all intermediate variables \(v_k\) part of this path do not relate to any other transformation \(g\) or transformation \(t\).

The third set of component pairs denotes those pairs \(c_i\) and \(c_j\) for which there exist a goal \((c_i, c'_i, V_i, S_i) \in \mathcal{G}\) and a transformation \((c_j, V_j, V'_j, S_j) \in \mathcal{T}\), such that there exists a path over the set of behavior dependencies between variables \(E_{\mathrm{v_b}}\) between \(v_i\) in \(V_i\) and a variable \(v_j\) in \(V'_j\) , such that all intermediate variables \(v_k\) part of this path do not relate to any other transformation \(g\) or transformation \(t\).

The fourth set is similar to the third set except here a flow in \(V_j\) is bound by an output flow in \(V'_i\).

For the decomposition-function-behavior view this set equals \(\{(\mathrm{c}_1, \mathrm{c}_2)\}\) since \((\mathrm{v}_{2}, \mathrm{v}_{3}) \in E_{\mathrm{v_b}}\) holds.

### Design dependencies¶

Design dependencies denote that designs of two components are dependent on each other but are not necessarily functionally connected. For example, two components may have to fit in a predefined space but do not exchange any (functional) flows. In this work, we refer to such dependencies as design dependencies. These dependencies are derived using variable set \(V_c\) of components and the set of design dependencies between variables \(\in E_\mathrm{v_d}\) Design dependency set \(E_\mathrm{c_d}\) is given by:

in which contains those pairs of components \((c, c )\) for which \(c_i\) and component \(c_j\) are not in each others trees and their exist two variables \(v_i \in V_i \cup P_i\) and \(v_j \in V_j \cup P_j\) which have a design dependency.

For the decomposition-relation-design view this set equals \(\{(\mathrm{c}_{1}, \mathrm{c}_{3}) (\mathrm{c}_{21}, \mathrm{c}_{22}), (\mathrm{c}_{21}, \mathrm{c}_{221}), (\mathrm{c}_{221}, \mathrm{c}_{222}) \}\) since the values of properties \(v_8\) and \(v_9\) are bound by one another due to design-rule \(q_1\).

### Coordination dependencies¶

Coordination dependencies denote the presence of a shared variable between two components of which the value has to be coordinated. The set of coordination dependencies between components \(E_\mathrm{c_c}\) is given by:

The this set contains pairs \((c_i,c_j)\) such that component \(c_i\) and component \(c_j\) are not in each others trees and the intersection of variable sets \(V_i \cup P_i\) and \(V_j \cup P_j\) is non-empty, i.e., \(c_i\) and \(c_i\) share one or more variables. In other words, this set contains components pairs that share variables.

For the example shown in the decomposition-relation-design view, this set equals \(\{(\mathrm{c}_{1}, \mathrm{c}_{2}), (\mathrm{c}_{1}, \mathrm{c}_{21})\}\) since these component pairs share variable \(v_8\). Note that component pair \((\mathrm{c}_{2}, \mathrm{c}_{21})\) is not part of this set as \(\mathrm{c}_{21} \in tree(\mathrm{c}_{2})\). We exclude these coordination dependencies as by definition the design of a parent component has to be coordinated with the design of its children.

## Dependencies between function specifications¶

Three types of dependencies between function specifications are derived: traceability dependencies, i.e., dependencies that show which function specifications specified at level \(k+1\) contribute to the fulfillment of function specifications specified at level \(k\), functional dependencies, i.e., dependencies that denote the function chains within the system, and behavior dependencies, i.e., dependencies that are derived from behavior specifications and denote logical dependencies.

### Traceability dependencies¶

Traceability dependencies are derived based on functional dependency paths between variables. All function specifications that contribute to the fulfillment of a transformation-specification \(t \in \mathcal{T}\), referred to as the children of \(t\) and are denoted by \(F(t) \subset \mathcal{F}\), have to be part of a function chain starting with an input variable \(v_i \in V_t\) and ending at an output variable \(v_j \in V_t'\). The set of all variables that are part of a path starting at an input variable \(v_i \in V_t\) and ending at an an output variable \(v_j \in V_t'\) over the scope of subject component \(c_t\) of transformation \(t\) is given by:

\(V(v_i, v_j, c_t)\) is used to compute the set of children of transformation functions \(F(t) \subset \mathcal{F}\). The set of children of an arbitrary transformation specification \(t\) is given by:

That is, a transformation specification \(t'\) is a child of transformation specification \(t\) if the subject component \(c_t'\) is a child of subject component \(c_t\), there exists a path from input variable \(v_i\) to output variable \(v_j\) over the scope of \(c_t\) such the the intersection of the set of input variables \(V_{t'}\) and the set of path variables \(V(v_i, v_j, c_t)\) and the intersection of output variables \(V_{t'}'\) and the set of path variables \(V(v_i, v_j, c_t)\) are non empty sets. In other words, \(t'\) is a child of transformation specification \(t\) if it is part of the function chain describing the transformation of an input variable \(v_i\) to an output variable \(v_j\).

A goal-specification \(g\) is a child of transformation-specification \(t\) if the subject component \(c_g\) and indirect object component \(c_g'\) are children of subject component \(c_t\) and the intersection of the set of goal variables \(V_g\) with \(V(v_i, v_j, c_t)\) is a non empty set. In other words, if one or more variables in \(V_g\) are path of an input-output path.

The function \(F(t)\) is used to determine the set of traceability dependencies \(E_\mathrm{f_t}\), which is given by:

For the decomposition-function view this set equals \(\{(\mathrm{t}_1, \mathrm{t}_2), (\mathrm{t}_1, \mathrm{g}_2), (\mathrm{t}_1, \mathrm{t}_3), (\mathrm{t}_3, \mathrm{t}_4), (\mathrm{t}_3, \mathrm{g}_4), (\mathrm{t}_3, \mathrm{t}_5) \}\). As such, transformation specifications \(\mathrm{t}_1\) and \(\mathrm{t}_3\) can be traced down to lower level function specifications.

Note that goal- and transformation-specifications can be part of multiple function chains describing multiple parent transformations. For example, the example figure below shows two transformations \(t_i\) and \(t_j\) with subject component \(c\) describing transformations from \(x\) to \(y\) and \(x\) to \(z\), respectively.

The three transformations \(t'\), \(t''\), and \(t'''\), where \(c_{t'}, c_{t''}, c_{t'''} \in C_c\), describe these transformations in more detail. \(t'\) describes a transformation from \(x\) to \(q\), \(t''\) describes a transformation from \(q\) to \(y\), and \(t'''\) describes a transformation from \(q\) to \(z\). So, \(t'\), \(t''\), and \(t'''\) describe the two transformation paths \(x \rightarrow q \rightarrow y\) and \(x \rightarrow q \rightarrow z\). As such, \(t' \in F(t_i)\) and \(t' \in F(t_j)\) both hold.

Therefore, \(t'\) has two parent functions (\(t_i\) and \(t_j\)) as it contributes to the fulfillment of both of these functions. In other words, the function hierarchies form directed acyclic graphs rather than tree structures.

```
flowchart LR
subgraph c
direction LR
x((x))
y((y))
z((z))
q((q))
x --> x'
q' --> q
q --> q''
q --> q'''
y'' --> y
z''' --> z
subgraph c'
direction LR
x'((x))
q'((q))
x' -.->|t'| q'
end
subgraph c''
direction LR
q''((q))
y''((y))
q'' -.->|t''| y''
end
subgraph c'''
direction LR
q'''((q))
z'''((z))
q''' -.->|t'''| z'''
end
x -.->|t<sub>i</sub>| y
x -.->|t<sub>j</sub>| z
end
```

Traceability dependency derivation for a component with subcomponents.

### Functional dependencies¶

Function dependencies denote the transfer and transformation of flows through the system. That is, they denote the function chains that are being fulfilled by the various components in the system. The set of all functional dependencies between functions \(E_\mathrm{f_f}\) is given by:

That is, transformation-specification \(t_j\) depends on transformation-specification \(t_i\) if the intersection of output variable set \(V_{t_i}'\) and input variable set \(V_{t_j}\) \(t_j\) is a non-empty set. Note that, no direct goal-specification dependencies \((g_i, g_j)\) are derived to avoid bypasses.

A goal-specification \(g\) depends on transformation-specification \(t\) if the intersection of output variable set \(V_t'\) with variable set \(V_g\) is not empty.

Similarly, transformation specification \(t\) depends on goal-specification \(g\) if the intersection of variable set \(V_g\) with input variable set \(V_t\) is not empty.

For the traceability dependencies example this set equals to \(\{ (\mathrm{g}_1, \mathrm{t}_1), (\mathrm{t}_1, \mathrm{g}_2), (\mathrm{g}_1, \mathrm{t}_2), (\mathrm{g}_2, \mathrm{t}_6), (\mathrm{t}_6, \mathrm{g}_5), (\mathrm{t}_2, \mathrm{g}_3), (\mathrm{g}_3, \mathrm{t}_3), (\mathrm{t}_3, \mathrm{g}_2), (\mathrm{g}_3, \mathrm{t}_4), (\mathrm{t}_4, \mathrm{g}_4), (\mathrm{g}_4, \mathrm{t}_5), (\mathrm{g}_5, \mathrm{g}_2)\}\). A such, at each level of the decomposition a closed function chain is found.

These new dependency derivations rules imply that only transformation-specifications that share the same subject component can have direct dependencies. Transformation-specifications that do not have the same subject component can only have indirect dependencies via goal-specifications. Goal-specifications can only have indirect dependencies via transformation-specifications.

As a consequence, a user must specify transformation functions at all decomposition levels to obtain complete function chains and function hierarchies. This differs from the current convention of only specifying transformation-specifications within leaf components (\(\{ c \mid C_c = \emptyset \}\)). This is a price we are willing to pay as it provides functional traceability and allows for completeness checks.

The set \(E_\mathrm{v_f}\) can be used to check the functional completeness of a specification. For example, assume we have a transformation specification \(t = (c_t, V_t, V_t', S_t)\) for which \(C_{c_t} \neq \emptyset\) then \(t\) is traceable if \(path(v_i, v_j, c_t) \neq \emptyset \ \forall v_i \in V_t, v_j \in V_t'\). That is, the transformations specified within leaf components of the tree with root \(c_t\) should describe a transformation path for all input variables \(v_i \in V_t\) to all output variables \(v_j \in V_t'\). The non-existence of such a path implies that the transformation \(t\) that should be fulfilled by \(c_t\) is not traceable to the subcomponents of \(c_t\). Thus, the specification is incomplete.

### Behavior dependencies¶

Behavior dependencies denote logical dependencies between functions. For example, the amount of torque delivered by an electric-motor may depend on a control-signal. Such a dependency is not physical but logical. The set of behavior dependencies between functions \(E_{\mathrm{f_b}}\) is given by:

which consists of three sets. The first set contains those goal- and transformation specification pairs \((g, t)\) for which there exists a path over the set of behavior dependencies between variables \(E_{\mathrm{v_b}}\) between a variable \(v_i \in V_g\) and a variable \(v_j \in V'_t\), such that all intermediate variables \(v_k\) part of this path do not relate to any other transformation \(g\) or transformation \(t\).

The second set contains those transformation-goal pairs for which there exists a path over the set of behavior dependencies between variables \(E_{\mathrm{v_b}}\) between a variable \(v_i \in V'_t\) and a variable \(v_j \in V_g\), such that all intermediate variables \(v_k\) part of this path do not relate to any other transformation \(g\) or transformation \(t\).

The third set contains those transformation-transformation pairs for which there exists a path over the set of behavior dependencies between variables \(E_{\mathrm{v_b}}\) between a variable \(v_i\) in \(V'_i\) and a variable \(v_j\) in \(V'_j\), such that all intermediate variables \(v_k\) part of this path do not relate to any other transformation \(g\) or transformation \(t\). In other words, the the output of \(t_j\) is bound by the output of \(t_i\).

For the decomposition-function-behavior view this set equals \(\{(\mathrm{g}_1, \mathrm{t}_1)\}\) since the value of \(\mathrm{v}_3\) depends on the value of \(\mathrm{v}_2\) as defined by \(\mathrm{h}_1\).

## Dependencies between behavior specifications¶

Two types of dependencies between behavior specifications are derived: behavior dependencies and coordination dependencies.

### Behavior dependencies¶

The set of dependencies between behaviors \(E_{\mathrm{h_b}}\) is given by:

That is, two behaviors \(h_i\) and \(h_j\) have a behavior dependency if at least one variable that is used within then-clauses \(S'_i\) of \(h_i\) is used within when-clauses \(S_j\) of \(h_j\). Such a dependency implies that the result of \(h_j\) is bounded by the result of \(h_i\).

### Coordination dependencies¶

Coordination dependencies \(E_{\mathrm{h_c}}\) are given by:

That is, two behaviors \(h_i\) and \(h_j\) have a coordination dependency if at least one variable that is used within a then-clauses \(S'_i\) of \(h_i\) is used within a then-clauses \(S'_j\) of \(h_j\). Such a dependency implies that both behaviors aim to set the values of a least one shared variable.

## Dependencies between need, design, and relation specifications¶

In this section the dependency sets between needs, design, and relation specifications are given. These dependencies set are not the main focus of this work, but provide opportunities to create other views on the network of dependencies if desired. Need dependency set \(E_\mathrm{n}\) is given by:

That is, two needs \(n_i\) and \(n_j\) have a dependency if the needs have the same subject \(s\).

Design specification dependency set \(E_\mathrm{d}\) is given by:

That is, two design specifications \(d_i\) and \(d_j\) have a dependency if they share at least one variable. For the example this implies that the design specifications \(d_i\) and \(d_j\) for which, \(q_1 \in Q_i\) and \(q_2 \in Q_j\) holds, have a dependency as they share variable \(v_9\).

Relation specification dependency set \(E_\mathrm{r}\) is given by:

That is, two relations \(r_i\) and \(r_i\) have a dependency if the intersection of the union of the output variables \(V_i'\) and undirected variables \(V_i''\) of relation \(r_i\) with the union of the input variables \(V_j\) and undirected variables \(V_j''\) of relation \(r_j\) is not empty. In other words, if an output or undirected variable of \(r_i\) is an input or undirected variable of \(r_j\), then \(r_j\) depends on \(r_i\). Note that if \(V''\) is empty for all \(r \in \mathcal{R}\) that all relation dependencies become directed.

## Mapping relations¶

In the previous sections component dependency sets \(E_\mathrm{c_f}\) and \(E_\mathrm{c_d}\), variable dependency sets \(E_\mathrm{v_f}\) and \(E_\mathrm{v_d}\), goal specification dependency set \(E_\mathrm{g}\), need dependency set \(E_\mathrm{n}\), transformation specification dependency set \(E_\mathrm{t}\), design specification set \(E_\mathrm{d}\), the relation specification dependency set \(E_\mathrm{r}\) are defined. These sets all denote dependencies between ESL elements of the same class, e.g., between components and between variables.

In this section, the focus is on connections between ESL elements of different classes. For example, a connection between a component and a variable or a connection between a goal specification and a design specification. These connections are referred to as mapping relations, which are undirected by definition. These relations are required to gain insight into the relations between the different ESL elements. For example, to gain insight into which flow and design variables a component relates to. The derivation of mapping relations is based on containment or shared variables. For instance, in the decomposition-function view goal-specification \(g_1\) contains component \(c_1\), component \(c_2\), and variable \(v_1\), as such, \(g_1\) has mapping relations with \(c_1\), \(c_2\), and \(v_1\). The set of mapping relations between all ESL elements \(M\) is given by:

In which the function \(V_\mathrm{h}(h)\) collects all variables used within a behavior requirement \(h\) and is given by:

\(M\) contains all mapping relations between all combinations of variables, components, needs, function-specifications, design-specifications, and relations. In the remainder of this article, we often refer to a subset of mapping relations between two distinct ESL element classes, which is given by:

where, \(\mathcal{X}\) and \(\mathcal{Y}\) are the element sets for which the subset of mapping relations is used. For example, \(M_{\mathcal{C}\mathcal{F}}\) is the subset of mapping relations between components \(\mathcal{C}\) and functions \(\mathcal{F}\).